This paper is concerned with the existence and stability of phase transition steady states to a quasi-linear hyperbolic–parabolic system of chemotactic aggregation, which was proposed in [Ambrosi, Bussolino and Preziosi, J. Theoret. Med. 6 (2005) 1–19; Gamba et al., Phys. Rev. Lett. 90 (2003) 118101.] to describe the coherent vascular network formation observed in vitro experiment. Considering the system in the half line ${\mathbb{R}}_{+}=(0,\infty )$ with Dirichlet boundary conditions, we first prove the existence and uniqueness of non-constant phase transition steady states under some structure conditions on the pressure function. Then we prove that this unique phase transition steady state is nonlinearly asymptotically stable against a small perturbation. We prove our results by the method of energy estimates, the technique of a priori assumption and a weighted Hardy-type inequality.

中文翻译：

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相变稳态到双曲线-抛物线系统建模血管网络的非线性稳定性

本文关注相变稳态到趋化聚集的准线性双曲线-抛物线系统的存在和稳定性，该系统在 [Ambrosi, Bussolino and Preziosi, J. Theoret.] 中提出。医学。6 (2005) 1-19；Gamba等人，物理学。莱特牧师。90 (2003) 118101.] 描述在体外实验中观察到的连贯的血管网络形成。考虑半线系统${\mathbb{电阻}}_{+}=(0,\infty )$利用狄利克雷边界条件，我们首先证明了压力函数上某些结构条件下非恒定相变稳态的存在性和唯一性。然后我们证明了这种独特的相变稳态对于小扰动是非线性渐近稳定的。我们通过能量估计方法、先验假设技术和加权哈代不等式证明了我们的结果。